where \(N=\#\lbrace [n]\bigcap [n-k]\rbrace \) , i.e., the number of 2-powers common to the base 2
expansions of \(n\)  and \(n-k\) ,
which he called the almost minimum zero estimate. Adelberg showed that the MZC occurs if and only if \(N=0\)  in (REF ), i.e., \(\nu _2(S(n,k))=\sigma _2(k)-\sigma _2(n)\) .
He also showed that his new estimate is always nonnegative, unlike De Wannemacker's estimate (1.3), which can be negative. If (REF ) is not sharp but (REF ) is sharp, we have an almost minimum zero case (AMZC).
In addition, he obtained certain “shifted" estimates and cases for \(p=2\) , which obviate the need for many inductions.
In this paper we generalize these concepts to arbitrary primes. In particular, the shifted cases give us insight into certain cases where \(\nu _p(S(n+1,k+1))=\nu _p(S(n,k))\)  and where \(\nu _p(s(n-1,k-1))=\nu _p(s(n,k))\) , continuing the generalization that was done in [1]} of what was originally a conjecture of Amdeberhan et al.; cf. [2]}. We give geometric interpretations of all the cases in terms of the Newton polygons of certain higher order Bernoulli polynomials.
